Let L be a linear map between linear spaces U and V, such that
L: U...
Let L be a linear map between linear spaces U and V, such that
L: U -> V and let l_{ij} be the matrix associated with L w.r.t
bases {u_i} and {v_i}. Show l_{ij} changes w.r.t a change of bases
(i.e u_i -> u'_i and v_j -> v'_j)
Let U and V be vector spaces, and let L(V,U) be the set of all
linear transformations from V to U. Let T_1 and T_2 be in
L(V,U),v be in V, and x a real number. Define
vector addition in L(V,U) by
(T_1+T_2)(v)=T_1(v)+T_2(v)
, and define scalar multiplication of linear maps as
(xT)(v)=xT(v). Show that under
these operations, L(V,U) is a vector space.
1. For a map f : V ?? W between vector spaces V and W to be a
linear map it must preserve the structure of V . What must one
verify to verify whether or not a map is linear?
2. For a map f : V ?? W between vector spaces to be an
isomorphism it must be a linear map and also have two further
properties. What are those two properties? As well as giving the
names...
Problem 3. An isometry between inner-product spaces V and W is a
linear
operator L in B (V ,W) that preserves norms and inner-products. If
x, y in V
and if L is an isometry, then we have <L(x),L(y)>_W = <x,
y>_V .
Suppose that V and W are both real, n-dimensional inner-product
spaces.
Thus the scalar field for both is R and both of them have a basis
consisting of
n elements. Show that V and W are isometric...
Let A ∈ L(U, V ) and B ∈ L(V, W). Assume that V is
finite-dimensional.
Let X be a 3 × 5 matrix and Y be a 5 × 2 matrix. What are the
largest and smallest possible ranks of X, Y, and XY? Give examples
of the matrix to support your answers
V and W are finite dimensional inner product spaces,T: V→W is a
linear map
1A: Give an example of a map T from R2 to itself (with the usual
inner product) such that〈Tv,v〉= 0 for every map.
1B: Suppose that V is a complex space. Show
that〈Tu,w〉=(1/4)(〈T(u+w),u+w〉−〈T(u−w),u−w〉)+(1/4)i(〈T(u+iw),u+iw〉−〈T(u−iw),u−iw〉
1C: Suppose T is a linear operator on a complex space such
that〈Tv,v〉= 0 for all v. Show that T= 0 (i.e. that Tv=0 for all
v).
Let X ∈ L(U, V ) and Y ∈ L(V, W). You may assume that V is
finite-dimensional.
1)Prove that dim(range Y) ≤ min(dim V, dim W). Explain the
corresponding result for matrices in terms of rank
2) If dim(range Y) = dim V, what can you conclude of Y? Give
some explanation
3) If dim(range Y) = dim W, what can you conclude of Y? Give
some explanation
V and W are finite dimensional inner product spaces,T:V→W is a
linear map, and∗represents the adjoint.
1A: Let n be a positive integer, and suppose that T is defined
on C^n (with the usual inner product) by T(z1,z2,...,zn) =
(0,z1,z2,...,zn−1). Give a formula for T*.
1B: Show that λ is an eigenvalue of T if and only if λ is an
eigenvalue of T*.
Let f : V mapped to W be a continuous function between two
topological spaces V and W, so that (by definition) the preimage
under f of every open set in W is open in V : Y is open in W
implies f^−1(Y ) = {x in V | f(x) in Y } is open in V. Prove that
the preimage under f of every closed set in W is closed in V . Feel
free to take V...
Questionnnnnnn
a. Let V and W be vector spaces and T : V → W a linear
transformation. If {T(v1), . . . T(vn)} is linearly independent in
W, show that {v1, . . . vn} is linearly independent in V .
b. Define similar matrices
c Let A1, A2 and A3 be n × n matrices. Show that if A1 is
similar to A2 and A2 is similar to A3, then A1 is similar to
A3.
d. Show that...